Mathematics – Differential Geometry
Scientific paper
2012-03-02
Mathematics
Differential Geometry
64 pages, 12 figures; Added a section regarding the intrinsic torsion of a generalized almost Hermitian manifold and provided
Scientific paper
SKT structures are closely related to Kahler structures, the difference being that in the Kahler case one has a complex structure which is parallel with respect to the Levi--Civita connection, while in the SKT the complex structure is parallel with respect to a metric connection with skew-symmetric and closed torsion, a concept which is a little more restrictive than asking that such connection has holonomy in U(n). The inclusion of the torsion, however leaves several of the usual arguments used in Kahler geometry without a direct counterpart. We use tools from generalized complex geometry to develop the theory of SKT manifolds. We develop Hodge theory on SKT manifolds and, more generally, on parallel Hermitian manifolds and prove that their twisted cohomology inherits a Z x Z_2-grading determined by the structure. We study Lie algebroids and differential operators associated to SKT structures and study the deformation theory of these structures. As applications we reobtain a result of Lubke and Teleman regarding the existence of SKT structures on the moduli space of instantons of a bundle over a complex surface and show that even though Kahler structures are not stable under deformations of the symplectic structure, small deformations are still SKT.
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